Beyond First-Order

Calculus 2 · Axiom Academy

A mass on a spring keeps coming back — and that one fact forces a second derivative, a second-order equation, and a solution that waves instead of grows. First-order equations grow or decay. This one swings. You already know first-order equations like y' = ky : things that grow or decay, and never turn around. But a mass bobbing on a spring does turn around — over and over, forever. To describe that you need a second derivative , and out of it falls a solution that no exponential can produce: a wave. Watch the mass let go and swing. As it moves, its height is recorded to the right — and the trace it leaves behind is a perfect . That sine wave is the solution of the equation we are about to build. No exponential ever comes back to where it started. A sine does, every period — that periodic return is what oscillation means. Why it keeps coming back: the restoring force Drag the mass up or down. The spring answers with a force F = -ky that always points back toward equilibrium — pull it down, the force pulls up; push it up, the force pushes down. By Newton's law that force is mass times acceleration, so acceleration depends on position : . Force and acceleration always carry the opposite sign of the position — that minus sign is the whole story. The pull is strongest when you're farthest out, zero at equilibrium. Why sine, and not an exponential?

This is the written version of the interactive lesson above. See the full Calculus 2 course.